Sepehr Assadi

On estimating maximum matching size in graph streams

We study the problem of estimating the maximum matching size in graphs whose edges are revealed in a streaming manner. We consider both insertion-only streams, which only contain edge insertions, and dynamic streams that allow both insertions and deletions of the edges, and present new upper and lower bound results for both cases. On the upper bound front, we show that an α-approximate estimate of the matching size can be computed in dynamic streams using O(n2/α4) space, and in insertion-only streams using O(n/α2)-space. These bounds respectively shave off a factor of α from the space necessary to compute an α-approximate matching (as opposed to only size), thus proving a non-trivial separation between approximate estimation and approximate computation of matchings in data streams. On the lower bound front, we prove that any α-approximation algorithm for estimating matching size in dynamic graph streams requires [EQUATION] bits of space, even if the underlying graph is both sparse and has arboricity bounded by O(α). We further improve our lower bound to Ω(n/α2) in the case of dense graphs. These results establish the first non-trivial streaming lower bounds for super-constant approximation of matching size. Furthermore, we present the first super-linear space lower bound for computing a (1 + e)-approximation of matching size even in insertion-only streams. In particular, we prove that a (1 + e)-approximation to matching size requires RS(n) · n1−O(e) space; here, RS(n) denotes the maximum number of edge-disjoint induced matchings of size Θ(n) in an n-vertex graph. It is a major open problem with far-reaching implications to determine the value of RS(n), and current results leave open the possibility that RS(n) may be as large as n/log n. Moreover, using the best known lower bounds for RS(n), our result already rules out any O(n · poly(log n/e))-space algorithm for (1 + e)-approximation of matchings. We also show how to avoid the dependency on the parameter RS(n) in proving lower bound for dynamic streams and present a near-optimal lower bound of n2−O(e) for (1 + e)-approximation in this model. Using a well-known connection between matching size and matrix rank, all our lower bounds also hold for the problem of estimating matrix rank. In particular our results imply a near-optimal n2−O(e) bit lower bound for (1 + e)-approximation of matrix ranks for dense matrices in dynamic streams, answering an open question of Li and Woodruff (STOC 2016).

Randomized Composable Coresets for Matching and Vertex Cover

A common approach for designing scalable algorithms for massive data sets is to distribute the computation across, say k, machines and process the data using limited communication between them. A particularly appealing framework here is the simultaneous communication model whereby each machine constructs a small representative summary of its own data and one obtains an approximate/exact solution from the union of the representative summaries. If the representative summaries needed for a problem are small, then this results in a communication-efficient and \emph{round-optimal} (requiring essentially no interaction between the machines) protocol. Some well-known examples of techniques for creating summaries include sampling, linear sketching, and composable coresets. These techniques have been successfully used to design communication efficient solutions for many fundamental graph problems. However, two prominent problems are notably absent from the list of successes, namely, the maximum matching problem and the minimum vertex cover problem. Indeed, it was shown recently that for both these problems, even achieving a modest approximation factor of \polylog{(n)} requires using representative summaries of size \widetilde{\Omega}(n^2) i.e. essentially no better summary exists than each machine simply sending its entire input graph. The main insight of our work is that the intractability of matching and vertex cover in the simultaneous communication model is inherently connected to an adversarial partitioning of the underlying graph across machines. We show that when the underlying graph is randomly partitioned across machines, both these problems admit \emph{randomized composable coresets} of size \widetilde{O}(n) that yield an \widetilde{O}(1)-approximate solution\footnote{Here and throughout the paper, we use \Ot(\cdot) notation to suppress \polylog{(n)} factors, where n is the number of vertices in the graph. In other words, a small subgraph of the input graph at each machine can be identified as its representative summary and the final answer then is obtained by simply running any maximum matching or minimum vertex cover algorithm on these combined subgraphs. This results in an Õ(1)-approximation simultaneous protocol for these problems with Õ(nk) total communication when the input is randomly partitioned across k machines. We also prove our results are optimal in a very strong sense: we not only rule out existence of smaller randomized composable coresets for these problems but in fact show that our \Ot(nk) bound for total communication is optimal for em any simultaneous communication protocol (i.e. not only for randomized coresets) for these two problems. Finally, by a standard application of composable coresets, our results also imply MapReduce algorithms with the same approximation guarantee in one or two rounds of communication, improving the previous best known round complexity for these problems.

Tight bounds for single-pass streaming complexity of the set cover problem

We resolve the space complexity of single-pass streaming algorithms for approximating the classic set cover problem. For finding an α-approximate set cover (for α= o(√n)) via a single-pass streaming algorithm, we show that Θ(mn/α) space is both sufficient and necessary (up to an O(logn) factor); here m denotes number of the sets and n denotes size of the universe. This provides a strong negative answer to the open question posed by Indyk (2015) regarding the possibility of having a single-pass algorithm with a small approximation factor that uses sub-linear space. We further study the problem of estimating the size of a minimum set cover (as opposed to finding the actual sets), and establish that an additional factor of α saving in the space is achievable in this case and that this is the best possible. In other words, we show that Θ(mn/α2) space is both sufficient and necessary (up to logarithmic factors) for estimating the size of a minimum set cover to within a factor of α. Our algorithm in fact works for the more general problem of estimating the optimal value of a covering integer program. On the other hand, our lower bound holds even for set cover instances where the sets are presented in a random order.

The Minimum Vulnerability Problem

We revisit the problem of finding

Tight Space-Approximation Tradeoff for the Multi-Pass Streaming Set Cover Problem

k

A Compile-Time Optimization Method for WCET Reduction in Real-Time Embedded Systems through Block Formation

k paths with a minimum number of shared edges between two vertices of a graph. An edge is called shared if it is used in more than one of the

Fast Convergence in the Double Oral Auction

k